OAI.ThreeState.poisson_supercritical

The theorem states that, for a three-state spin model with spins in {0,1,2}, take a real d > 1 and a parameter λ that is admissible, meaning -1/2 ≤ λ ≤ 1. Each spin is passed through a noisy channel that keeps it with probability (1+2λ)/3 and changes it to each other particular spin with probability (1-λ)/3. Observations are built recursively: at depth 0 one sees the spin itself, and at depth n+1 the root draws a Poisson(d) number k of children, passes each of k independent child spins (each obtained from the root spin through the channel) through the depth-n observation procedure, and records the resulting unordered multiset of k depth-n observations. The advantage at depth n is the expected total-variation distance between the posterior on the root spin, given the observation under a uniform prior, and the uniform distribution on the three spins, namely the average over observations y of half the sum over spins i of |posterior(i | y) - 1/3|. Assuming the supercritical condition 1 < d·λ², the theorem concludes that this Poisson advantage sequence reconstructs, meaning it converges as n tends to infinity to some limit L that is strictly positive. The statement is admitted without proof in the source.